We consider the case \(p=1\)  and \(q=\infty \)  as in [1]} and  [2]},
and aim at bounding \( \left\Vert \leavevmode {\color {red}d_{\tilde{\pi }}} - \leavevmode {\color {blue}d_{\pi }}\right\Vert _1\)  and \(\left\Vert A_{\pi }^{\tilde{\pi }}\right\Vert _\infty \) .
